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TESIS DOCTORAL

New Developments in the Partial Control

of Chaotic Systems

Autor:

Rub´

en Cape´

ans Rivas

Directores:

Miguel ´Angel Fern´andez Sanju´an

Juan Sabuco

Programa de Doctorado en Ciencias Escuela Internacional de Doctorado

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Agradecimientos

Esta tesis es el resultado de a˜nos de investigaci´on y continuo aprendizaje. Un camino que sin duda ha estado influenciado por muchas personas que con su experiencia, consejos y ayuda han hecho posible este trabajo. En primer lugar quiero expresar mi agradecimiento a mi director de tesis Miguel ´Angel Fern´andez Sanju´an, catedr´atico de F´ısica de la Universidad Rey Juan Carlos, por ofrecerme esta primera oportunidad y depositar en m´ı la confianza para acometer este proyecto investigador en un campo tan apasionante. He de agradecerle adem´as su continuo apoyo, as´ı como el tiempo dedicado a la supervisi´on y mejora de todos los trabajos que dieron lugar a esta tesis.

A mi codirector Juan Sabuco Larrosa, cuya ayuda y gu´ıa en esta tesis ha sido fundamental. Este trabajo es en cierto modo, una continuaci´on de su trabajo doc-toral acerca del m´etodo de control parcial de sistemas ca´oticos. Han sido muchas y muy enriquecedoras las discusiones, ideas, y revisiones que Juan ha aportado a este trabajo.

A mis profesores de la Universidad de Santiago de Compostela, que me ense˜naron muchas de las herramientas usadas durante esta tesis. Especial importancia para m´ı tienen algunas de las asignaturas cursadas en el ´ultimo curso de licenciatura donde descubr´ı el mundo de la f´ısica no lineal y de la complejidad. Enseguida encontr´e fascinante esta rama de la F´ısica y sin duda me marc´o conceptualmente hasta el punto de querer continuar aprendiendo e iniciarme en la investigaci´on.

A Jes´us Seoane por su disponibilidad para solucionar cualquier duda o problema, as´ı como su eterno sentido del humor y por supuesto, a mis compa˜neros del grupo de Din´amica No lineal de la URJC, por ser magn´ıficos amigos y crear entre todos un ambiente de trabajo acogedor a la vez que estimulante.

Durante esta tesis se ha realizado una estancia en el extranjero. Quiero agradecer al Prof. Balakumar Balachandran del Departamento de Ingenier´ıa Mec´anica de la Universidad de Maryland por acogerme y poner a mi disposici´on el material de su Laboratorio. Gracias tambi´en a Vipin y Celeste por su acogida y ayuda durante esta estancia. En este mismo periodo, he podido conversar con el profesor James A. Yorke al cual agradezco su tiempo y buenos consejos, as´ı como sus contribuciones a algunos de los trabajos que figuran en esta tesis.

Quiero agradecer tambi´en a mi familia, que aunque no entiendan muy bien lo que hago, siempre han mostrado su inter´es y apoyo. A mis padres Jos´e y Maricarmen, por su confianza y paciencia, y por hacer posible que haya llegado hasta aqu´ı. A mis hermanos Pablo y Luc´ıa por ser parte de esta aventura y a la vez permitirme

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ser parte de la suya.

Por ´ultimo, el trabajo de investigaci´on que ha dado lugar a esta tesis doctoral se ha beneficiado de los proyectos de investigaci´on FIS2009-098981 del Ministerio de Ciencia e Innovaci´on y FIS2013-40653-P del Ministerio de Econom´ıa y Competitivi-dad. Tambi´en se ha beneficiado del proyecto de investigaci´on FIS2016-76883-P de la Agencia Espa˜nola de Investigaci´on (AEI) y el Fondo Europeo de Desarrollo Regional (FEDER). Igualmente se ha beneficiado por parte de la Universidad Rey Juan Carlos de una beca de ayuda a la movilidad para la realizaci´on de estancias predoctorales en Universidades y Centros de Investigaci´on extranjeros, dentro del Programa Propio de Investigaci´on, que tuvo lugar durante el a˜no 2016 en la Universidad de Maryland.

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Preface

This thesis has been developed during the past years in the Research Group on Nonlinear Dynamics, Chaos and Complex Systems of the URJC. All this work is devoted to new developments of the partial control method. The main goal of this technique is to control chaotic dynamics with escapes and affected by external disturbances. This thesis is organized as follows.

Chapter 1. Introduction

This chapter is a brief introduction to the main topics of our work. We describe the first steps of chaos theory and how the need of control arose in that field. Then we analize the main features of transient chaotic behaviour and the first attempts to control it. Finally, we show the evolution of the partial control method from the first ideas until the point this thesis was started.

Chapter 2. Description of the partial control method

The partial control method is used under different approaches along this thesis. In this chapter a general introduction to this method is given. The motivation to apply this method and the main dynamical conditions to apply it are presented. An algorithm to compute safe sets and how this set is used to control the system, is briefly described.

Chapter 3. Partial control to avoid a species extinction

In this chapter we present the first application of the partial control method in this thesis. Here, we have worked with an ecological model that describes the interaction between 3-species: resources, consumers and predators. The interest of this model lies in the fact that, for a choice of parameters, transient chaos involves the extinction of one of the species. Taking into account that the system is affected by external disturbances, we implement the partial control with the goal of avoiding the extinction.

Chapter 4. Controlling chaos in the Lorenz system

The Lorenz system is one of the most well-known systems in Nonlinear Dynamics. This makes it an excellent candidate to show how the partial control method can be applied in different ways depending on our requirements. For a certain choice

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of parameters, trajectories of this system eventually converge to two fixed points attractors via transient chaos. In order to avoid this escape, we describe three different ways based on building maps of one, two and three dimensions, respectively. Pros and cons of each one are analized, and for the first time a three-dimensional safe set is shown.

Chapter 5. A different application of partial control

In all the previous works, the computed safe set were used to keep the trajectories in the region of interest. Here we consider a new application of the safe set. Without any extra computation, we show in this chapter how this set can be also used to accelerate the escape of the trajectories if necessary. This fact, allows the controller a great flexibility to avoid or force the escape when it is required.

Chapter 6. When disturbance affects a parameter

Random maps are discrete dynamical systems where one or several of their pa-rameters vary randomly at every iteration. It is possible to find in these maps a transient chaotic behaviour, however few methodologies have been proposed to con-trol them. Here, we propose an extension of the partial concon-trol method, that we call parametric partial control. To do that, we consider the scenario where the distur-bances and the control terms are affecting directly some parameter of the system. To illustrate how the method works, we have applied it to three paradigmatic models in Nonlinear Dynamics, the logistic map, the H´enon map and the Duffing oscillator.

Chapter 7. Controlling time-delay coordinate maps

Delay-coordinate maps are a family of discrete maps where the dynamics have certain dependence on past states of the system. We consider these maps specially relevant because they can appear in the delay reconstruction technique of time series from experimental data. The main obstacle of these maps is that only the present state of the system can be modified. In this chapter, we study the convenience of the application of partial control under this constraint. To do that, a modified version of the partial control method is presented and some examples are illustrated. For the first time, it is treated a system that exhibits Hamiltonian chaos, and also a system that presents hyperchaos.

Chapter 8. A new approach: the safety functions

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special emphasis in the time series example. We believe that this work will open a door to new and stimulating applications in the field of control of chaotic systems.

Chapter 9. Discussion

A brief overview of the main results of this thesis and the possible research lines for a future work, is given in this chapter.

Chapter 10. Conclusions

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Contents

1 Introduction 1

1.1 A brief history of chaos . . . 1

1.2 Chaos and control . . . 2

1.3 Transient chaos and control . . . 3

1.4 Evolution of partial control . . . 6

2 Description of the partial control method 11 2.1 Introduction . . . 11

2.2 The partial control method . . . 11

3 Partial control to avoid a species extinction 17 3.1 Partial control to avoid a species extinction . . . 17

3.2 The partial control method implies smaller controls . . . 22

3.3 Conclusions . . . 24

4 Controlling chaos in the Lorenz system 27 4.1 Partial control to avoid the fixed point attractors in the Lorenz system . . . 27

4.1.1 The 1D map . . . 28

4.1.2 The 2D map . . . 29

4.1.3 The 3D map . . . 30

4.2 Conclusions . . . 35

5 A different application of partial control 37 5.1 Safe set to avoid the escape or force it . . . 37

5.1.1 The logistic map . . . 38

5.1.2 The H´enon Map . . . 40

5.2 Conclusions . . . 42

6 When the disturbance affects a parameter 43 6.1 Partial control applied on parameters . . . 43

6.1.1 The logistic map . . . 45

6.1.2 The H´enon map . . . 46

6.1.3 The Duffing oscillator . . . 47

6.1.4 Controlling more parameters . . . 48

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x Contents

6.2 Conclusions . . . 48

7 Controlling time-delay coordinate maps 51 7.1 The partial control applied to time-delay coordinates maps . . . 51

7.1.1 The two-dimensional cubic map . . . 54

7.1.2 The standard map . . . 56

7.1.3 The 3-dimensional hyperchaotic H´enon map . . . 57

7.2 Conclusions . . . 59

8 A new approach: the safety functions 61 8.1 Introduction . . . 61

8.2 Extending the partial control method . . . 61

8.2.1 Computing the function Uk in absence of disturbances. 63 8.2.2 Computing the function Uk in presence of disturbances 66 8.3 The safety function U∞ and the safe sets. . . 69

8.3.1 Application to the tent map affected by asymmetric disturbances . . . 70

8.3.2 Application to the H´enon map . . . 72

8.3.3 Application to a time series from an ecological system. 73 8.4 Conclusions . . . 77

9 Discussion 81 9.0.1 Considerations about the method . . . 82

9.0.2 Future work . . . 83

10 Conclusions 85 Bibliography 87 References . . . 87

Resumen de la tesis en castellano 91 Introducci´on . . . 91

Antecedentes del control parcial . . . 91

Avances en el m´etodo de control parcial . . . 92

Metodolog´ıa . . . 94

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Chapter 1

Introduction

1.1

A brief history of chaos

In the middle of the 17th century a great revolution in mathematics came up with the development of Calculus due to Isaac Newton and Gottfried Leibniz. One of the first and more significant applications was the development of Celestial Mechanics. With the knowledge of the gravitational interaction between the masses of celestial objects and its current position, it was possible to predict the future positions of the bodies, simply by using a set of differential equations.

Soon, this method was successfully applied to many other physical systems, creat-ing a certain “euphoria” that is well summarized in the mathematician Pierre-Simon Laplace thought, who believes that given the initial state and the physical laws gov-erning a system, its evolution can be perfectly predicted.

In the middle of the 19th century, it became clear that the motion of gases was far more complex to calculate than that of planets. The amount of calculations required for these systems made useless the mechanical approaches, thereby leading James Clerk Maxwell and Ludwig Boltzmann to create statistical physics. This was one of the first warnings about the limitation of predictability in physics, but still persisted the idea that, with the sufficient knowledge and calculus power, even those systems could be perfectly predicted.

In 20th century, one of the biggest revolutions appears in physics, Quantum Mechanics. The uncertainty principle stated by Werner Heisenberg, was the first big conceptual obstacle against the predictability principle. The position and velocity of an object cannot be, even in theory, exactly measured. This physical limitation involves that the Laplace presumption is not physically realizable. However the effects of the uncertainty principle are so small in ordinary size objects, that the Laplace statement remains still applicable in the sense that approximate causes follow approximate effects.

Few decades ago, at the end of 19th century, the French mathematician Henri Poincar´e, studying the stability of the solar system, showed mathematically that even low-dimensional systems like the famous 3-body problem, can exhibit a highly complex dynamical behaviour. He realized that in some nonlinear systems, even

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2 Chapter 1. Introduction

small deviations in the initial conditions could produce enormous differences in the final state of the system. Long term predictions become impossible, burying for-ever the Laplace dream. Later, in 1892, Poincar´e published his major work Les M´ethodes nouvelles de la M´ecanique C´eleste where new fundamental tools in Non-linear Dynamics were introduced. He is considered one of the founding fathers of chaos theory.

The enthusiasm generated by the discovery of Quantum Mechanics and Special Relativity in physics, provoked that the studies of Poincar´e and others like George D. Birkhoff, Eberhard Hopf, or Andrey Kolmogorov did not receive much attention in the physics community. The complexity that often comes out from the study of nonlinear systems made it very difficult to achieve substantial results in this field. It was only with the advent of computers, and the access to a huge amount of calcula-tions, when serious studies of nonlinear systems were possible. In this sense, one of the first works was made by the American meteorologist Edward Lorenz in 1963. He recognized the unpredictability of the dynamical behaviour in connection with the numerical solution of the model named after him. He also observed that his simple model of three ordinary differential equations presented non-periodic solutions, and for some value parameters, the trajectories are attracted to an strange topological object, not a surface, neither a volume, that was found to be fractal (Mandelbrot, 1982). This kind of new motion was first named chaos by the American mathemati-cian James Yorke in 1975.

1.2

Chaos and control

The word chaos itself may seem confusing if one interprets it with the colloquial meaning of “lack of order”. However deterministic chaotic systems are quite ordered and even predictable on short-time scales. The goal of modern dynamicists is to find the hidden order in the apparent chaos. In this sense important contributions have been made by J. A. Yorke, J. P. Eckmann, P. Grassberger, C. Grebogi, M. H´enon, P. Holmes, E. Ott, O. R¨ossler, D. Ruelle, Y. Sinai, and S. Smale among others. They have shown, roughly speaking, that chaos is the consequence of infinite local instabilities (over short times nearby states that move away from each other), but with stable behavior over long times.

Nowadays, chaos still admits several technical definitions, however the common underlying ideas (see Ref. [1]) can be described in a few variables. Without going into details the main features and associated measures to describe chaotic motion are:

• Irregular in time, aperiodicity →positive topological entropy.

• Sensitivity to initial conditions (unpredictable in the long term) → positive average Lyapunov exponent.

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1.3. Transient chaos and control 3

Any of these properties may help in deciding whether a system is chaotic or not. With the introduction of computers, chaotic dynamics was found in many dif-ferent fields like meteorology, geophysics, plasmas and lasers, electronic circuits, acoustics among others. The high irregular and unpredictable motion associated with chaos, can make desirable in some situations to suppress the chaotic behaviour and make the system to behave in a regular and predictable manner. At the be-ginning, some physicists thought that this control task was impossible to achieve. Considering that for a chaotic system any small variation in the system’s state is followed by an exponentially growing variation of the motion, it was thought that any attempt to guide a chaotic system by using small perturbations would just lead it to other chaotic and uncontrolled motions.

However, this was the wrong point of view. If we have a system that presents sensitive dependence to initial conditions, a small but accurately chosen perturbation might induce a huge change in its dynamics. Thus, the exponential divergence of the initial conditions induced by chaos can be considered an advantage. In 1990 Edward Ott, Celso Grebogi and James A. Yorke addressed this question. In a seminal paper they proposed the stabilization of some of the infinite unstable periodic orbits embedded in the chaotic attractor by applying small temporal perturbations to an accessible parameter of the system (OGY method, [2]).

The OGY approach requires quite a lot of knowledge about the orbit, including its position and stability properties that are not always straightforward to find in a real-life situation. A different approach that lacks the above shortcomings but pursue the same goal was proposed by the Lithuanian physicist Kestutis Pyragas [3]. He proposed a delayed feedback control in the form of a signal that is proportional to the difference between the current system state and its state a period T ago. This approach is especially appealing for experimentalists, since one does not need to know anything about the target orbit beyond its period T.

1.3

Transient chaos and control

Under certain circumstances chaotic behaviour is only of finite duration, i.e. the complexity and unpredictability of the motion can be observed over a finite time interval. This type of chaos is called transient chaos.

Chaotic transients emerge when, due to the change of some parameter of the system, trajectories can escape from the chaotic region. When the parameter reaches this critical value, it is referred to as a crisis. To explain the main mechanism of a crisis it is necessary to introduce the concept of basin of attraction. The set of initial conditions leading to some asymptotic final state is called basin of attraction, and its geometry is specially related with the kind of motion present in the system. When more than one attractor is present, there is a boundary between the basins, and this can be a smooth curve or even a fractal curve. The main mechanisms [1] that lead to a chaotic crisis are:

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4 Chapter 1. Introduction

Figure 1.1. Transient chaos. The figure illustrates the chaotic transient behavior of a trajectory. Starting in the red point, the trajectory falls in the chaotic region where it remains for a while. After a finite amount of time, the trajectory escapes from the chaotic region towards an external attractor (blue fixed point). The transient chaotic behavior is due to the presence of a chaotic saddle in phase space. This invariant set is a non-attractive chaotic set and this is the reason why the trajectory eventually escapes. The goal of applying control is to sustain the chaotic behavior forever, avoiding the escape of the trajectories.

The chaotic attractor ceases to exist and it is converted into a chaotic saddle, which is an invariant fractal set in the phase space.

• Internal crisis→the chaotic attractor suddenly enlarges and a saddle is merged with a small size attractor.

• Basin boundary metamorphosis→the basin boundary changes its fractal char-acter: a hyperbolic point on the boundary becomes part of the chaotic saddle.

In comparison with permanent chaos, the basic new feature here is the finite lifetime of chaos. Almost all initial conditions in the chaotic region, escape after a chaotic transient. There is an exception, a few set of points (a Cantor-like set that is known as the chaotic saddle) that never escapes, but any minimal deviation from this set lead the trajectory to the external attractor. If individual lifetimes for different initial conditions are computed, the ensemble is fractal. This feature together with the fractal basins of attraction represents the main fingerprints to identify transient chaos.

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disap-1.3. Transient chaos and control 5

Figure 1.2. Loss regions in the Dhamala and Lai control method. The figure represents a one-dimensional map that presents transient chaos. Orbits behave chaotic in the interval [0.58,0.73]. When the orbit passes through the segment b−c, escapes from that interval. The central red set represents the original escape regions. However, when noise is present orbits can also escape through the right escape region (also in red). The black sets represent the two target regions (or loss regions) used to control the trajectories with the aim of avoiding the escape. We colored the regions with different thickness just to help the visualization.

pearance of chaos may be the signal of pathological behavior [7]. In all these cases, chaos is a desirable property that is worth preserving.

With the aim of sustaining the chaotic behaviour in the case of transient chaos, different control methods were proposed. The most important are the methods proposed by Yang et al. ([7], 1995), Schwartz and Triandaf ([8], 1996) and Dhamala and Lai ([9], 1999). Although the control scheme is different in each case, all of them are based on identifying some“loss regions” of the dynamics. Chaotic trajectories sooner or later pass trough these regions to then escape to an external attractor. The idea of these methods is to use the “loss regions” or pre-images of them as a control regions. Every time a trajectory pass through these regions, a suitable control is applied to re-inject the trajectory to the nearest chaotic region. (see Fig 1.2).

These methods work well in ideal conditions, however practical experimenta-tion always involve approximaexperimenta-tions and external factors interfering with the system. Even if the deviations from the experiment and the theoretical model are very small, the action of the unstable chaotic dynamics makes that these deviations grow ex-ponentially. For example, in the control methods cited before, if the choice of “loss regions” is not made carefully one might kick the dynamics into faster escaping regions, having as a consequence the increase of the frequency of perturbations re-quired to maintain chaos.

The other important problem is the unavoidable presence of some amount of noise in all real experiments. Even if this amount of noise is small, it should be taken into account since the chaotic motion is an error amplifier, and small deviations at the beginning can ruin even the best control strategies.

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6 Chapter 1. Introduction

new paths. For example, in the case of control by means of the loss regions, the presence of noise makes it possible that some trajectories leave the chaotic region without going trough the loss region, and making the control strategy to fail (see Fig. 1.2). However, there are other sources of deviations that should be taken into account. For example, mismatches in the mathematical modelling of the system or imprecision in the application of control must also be considered when a robust control strategy is designed. For all these reasons the partial control approach was conceived.

1.4

Evolution of partial control

The partial control method was proposed with the aim of sustaining the transient chaotic behaviour indefinitely in certain regionQof phase space, avoiding the unde-sirable escape of the trajectories. With a similar goal, different control methods have been proposed in the literature ([8, 9, 10, 11]). However, these methods differ from the partial control method in that they have been mainly designed to be applied in deterministic systems, while partial control is a robust method able to deal with random disturbances affecting the systems. The more remarkable feature of the partial control is the ability to keep a control smaller than the distur-bances. This counterintuitive and surprising result is possible due to the presence of the chaotic saddle in the phase space which is responsible of the transient chaos. The partial control method benefits from the fractal structure of the chaotic saddle, to reduce the impact of the disturbances and at the same time, to enhance the effect of the control applied.

The development of the partial control method begins with the Yorke’s game of survival ([12], 2004). This game was based on the tent map dynamics for a parameter value where transient chaos exists. In the game some external disturbance is present and the goal was to design a control strategy to remain in the chaotic region. The work revealed that some pre-images of the middle point of the map can be used as safe points. Trajectories forced to pass through these points can counteract the effect of disturbances with a smaller control effort.

This strategy was later extended to two-dimensional maps by Zambrano et al. ([13, 14]) in 2008 and 2009. They made use of the fact that typically chaotic sad-dles arise due to the existence of a horseshoe map in phase space. The particular geometrical action of this map may involve the existence of transient chaos on the system considered. However, they showed that precisely this geometrical action also implied the existence of certain sets, the safe sets, that can be used to keep the tra-jectories close to the chaotic saddle (see Fig 1.3). Also in these works they proposed a generic procedure to obtain these safe sets in every system with a horseshoe map.

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1.4. Evolution of partial control 7

Figure 1.3. Sets in the H´enon map. This figure corresponds to the H´enon map where the central square (dotted line) maps in the horseshoe region. Note that some points map outside the initial square and therefore escape from it. Taking the central line (thick black line) it is possible to compute the preimagesS1(black),S2(grey) andS3(silver) consisting

on 1, 2, 4 and 8 vertical curves, respectively. These lines can be used to control the orbits and avoid their escape from the initial square.

sets it has been proven that trajectories can be kept inQ, even with a control smaller than the external disturbances. (See Fig. 1.4)

The next step was to generalize this to an arbitrary set. In 2012 Sabuco et al. ([16, 17]) proposed a new approach to generalize the search of safe sets in systems of

arbitrary dimension that exhibit transient chaos. The algorithm that was proposed was able to compute the safe sets for a specified region in phase space, given the map, the maximum disturbance value, and the maximum allowed control. They called it the Sculpting Algorithm, because literally the algorithm sculpts the safe sets, discarding the points of the initial region that can not be controlled. An example of the application in the Duffing oscillator is shown in Fig 1.5.

This thesis started from this point showing how the partial control technique can be applied to different models. In Chapter 3, an experimental ecological model is treated. For certain parameters one of the species of the model gets extinct after a chaotic transient. We will show how the application of partial control is able to avoid the extinction. In Chapter 4, the method is applied to the well-known Lorenz system. Among other results we compute for the first time, 3-dimensional safe sets.

A new application of the safe sets is studied in Chapter 5. Instead of using the safe sets only to sustain the trajectories in some region, we use them also to accelerate the escapes of the trajectories. This dual application of the safe sets, give the controller a great flexibility since the trajectories can be kept chaotic or force them to escape when it is needed.

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8 Chapter 1. Introduction

a)

b)

c)

Figure 1.4. Safe sets from escape times sets in the H´enon map. (a)The escape time set, i.e., the set of points that escape from Q after 3 or more iterations. It consists of four pairs of strips. We have also plotted in black the images of those points under one iteration. Due to the fact that the images of the strips do not lay between the pairs of strips, it is impossible to use these images to keep trajectories inside Q with a control smaller than the disturbances . (b) Safe set after sculpting the escape sets. Now the images fall inside the bands. (c) A zoom of escape sets. Independently from the disturbance deviation

ξ a smaller control ucan put the trajectory back on the safe set.

of the system a delta time ago).

Finally, in Chapter 8, a new and more general approach of partial control is presented. Instead of working with safe sets, we introduce a new tool, the safety functionsU∞. This function is defined in the regionQ, and represents the minimum

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1.4. Evolution of partial control 9

(a) basins of attraction (b) Periodic attractors

(c) Initial region Q (d) Safe set

Figure 1.5. Periodic attractors of the Duffing oscillator. (a) In this figure it is drawn the complex structure of the phase space for the Duffing oscillator ¨x+0.15 ˙x−x+x3=

0.245 sint. Three different basins of attraction (magenta, blue and green) are presented in this system. The invariant unstable manifold associated to the chaotic saddle appears in yellow. (b) This figure shows the periodic attractors: two period-1 attractors and one period-3 attractor. We also show with circles of radius 0.2 the region of the phase space that we want to avoid, whatever the disturbances. (c) We use a grid of 6000×6000 points in the square [−2,2]×[−2,2] as our initial set, but removing the zones that we want to avoid, that is the circles. Applying the Sculpting Algorithm over several iterations, we will obtain the desired safe set. We let ξ0 = 0.08 be the maximum size of the vector

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Chapter 2

Description of the partial

control method

2.1

Introduction

The transient chaotic behavior is caused by the presence of a chaotic saddle in phase space. The main mechanism to create a chaotic saddle is when a chaotic attractor collides with the boundary of its own basin of attraction, causing a boundary crisis. In contrast to a chaotic attractor that possesses a fractal structure only in the stable direction, the chaotic saddle is a nonattractive invariant set that is fractal in both, the stable and unstable directions. Due to the fractal structure in the unstable direction, infinite holes arise along the unstable manifold of the chaotic saddle. A trajectory that is initially attracted along the stable direction for some finite amount of time, eventually escapes through one of the gaps present in the unstable direction. These escapes allow the trajectories to reach other regions of the phase space, involving catastrophic consequences for the system in some cases. For example in a thermal combustor model [5], the crisis leads to the flame-out making the device useless. Also, in the McCann-Yodzis ecological model [18] the crisis conducts irreversibly to the extinction of one of the species.

With the aim of avoiding these undesirable escapes, even when an external dis-turbance is affecting the chaotic dynamics, the partial control method was proposed. This control method, and some variations of it, are used along this thesis. In this chapter a general introduction of the partial control method is given.

2.2

The partial control method

The partial control method is a recently developed control strategy ([16, 17]) for preventing escapes associated with a transient chaotic behaviour. It is particularly appropriate when it is desirable to keep the magnitude of the control small in a sys-tem affected by external disturbances [19, 20]. Since in many experimental syssys-tems, the presence of disturbances may be unavoidable, a robust control strategy must

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12 Chapter 2. Description of the partial control method

take it into account, especially when it is necessary to keep the control as small as possible.

Methods that perform well in systems in absence of disturbances can fail dra-matically when disturbances appear. For this reason, it is necessary to consider a term, that we call disturbance, that encloses all the uncertainty affecting the dy-namics of the system, like modelling mismatches, finite precision in the measure of initial conditions or even systematic or random external disturbances. In most cases the amplitude of these disturbances can be limited, so we consider in the method bounded disturbances. On the other hand, the control available is usually limited by the experimental conditions. For this reason, we also consider in this scheme that the control available in the most real scenarios, is bounded.

The intrinsic instability of the chaotic saddle together with the action of distur-bance creates a difficult scenario where keeping the control small might seem not realistic. However it is possible to keep the trajectories close to the chaotic saddle taking advantage of the horseshoe map present in the phase space that produces the chaotic saddle. This geometrical action implies the existence of certain sets called safe sets that lie in the vicinity of the chaotic saddle. These sets are used to keep the trajectories controlled (close to the chaotic saddle).

To find these sets, we consider that in the region Qof the phase space where the transient chaos is located, the dynamics can be described with the map:

qn+1=f(qn) +ξn+un, with |ξn| ≤ξ0, |un| ≤u0.

Here qn ∈ Rn represents certain phase state of the system, and we assume that the mapf acts on a regionQlike a horseshoe map ([17]). The disturbanceξaffecting the map is considered to be bounded so that |ξn| ≤ ξ0. The control term u is also

limited so that |un| ≤u0.

Without the action of a disturbance and a control, nearly all trajectories inside

Q (except a zero measure set) escape from it after some iterations. However if disturbances are present, all trajectories eventually escape.

Under this control scheme, the safe set (represented by Q∞) is a subset of Q,

such that for all q ∈Q∞, the trajectoryqn+1 =f(qn) +ξn+un stay in Q∞ forever

(see Fig. 2.1). The control un is chosen at each iteration, with the knowledge of

f(qn) +ξn, and applied to place the trajectory again in the set Q∞. We say that

trajectories found under these conditions areadmissible trajectories.

The set Q∞ can be directly computed following an iterative process. Starting

with the set Q represented by a grid stored in the computer, the algorithm takes initially the setQ0 =Q. Then, given the grid pointq∈Q0 and the disturbed image f(q) +ξ, the algorithm checks whether exist a suitable control |u| ≤ u0 such that f(q) +ξ+ufalls again in Q0. If all possible disturbed images f(q) +ξcorresponding

toqare controllable, then the pointqis conserved. If not, it is removed. Proceeding similarly with all points of Q0 a new set Q1 ⊂Q0 is obtained.

However we do not know yet if Q1 is a safe set. To do that, it is necessary to

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2.2. The partial control method 13

Figure 2.1. Dynamics in Q0 and Q∞. The left side shows an example of a region Q0 (in blue) in which we want to keep the dynamics described by qn+1=f(qn) +ξn+un.

We say that |ξn| ≤ξ0 is a bounded disturbance affecting the map, andun is the control

chosen so that qn+1 is again in Q0. To apply the control, the controller only needs to

measure the state of the disturbed system, that is [f(qn) +ξn]. The knowledge of f(qn)

or ξnindividually is not required. The right side of the figure, shows the region Q∞⊂Q0

(in blue), called the safe set, where each xn ∈ Q∞ has xn+1 ∈ Q∞ for some control

|un| ≤ u0, which is chosen depending on ξn. Notice that the removed region does not

satisfy|un| ≤u0.

a result, a smaller set Q2 ⊂ Q1 ⊂ Q0 is obtained. This process is repeated until

it converges, in which case Q∞ is found. This set is known as the safe set. All

the disturbed images f(q) +ξ corresponding to the safe set, can be put them back (with control |u| ≤u0) again in the safe set. Due to this, the safe set is a positively

invariant set [21, 22, 23, 24, 25]. The surprising result, and the main fingerprint of this method is the existence of safe sets with control values 0> u0 > ξ0. This means

that the action of a disturbance to distort the dynamics can be counteracted with a smaller effort of control.

Based on the process presented above, it has been developed an algorithm called the Sculpting Algorithm [16], to automatically compute the successive regions

Qn until the safe set is finally found. We illustrate the procedure of finding the safe set in Fig. 2.2. We fix the bound u0 and ξ0 and the region Q0 = Q. The ith step

can be summarized as follows:

1. Morphological dilation of the setQibyu0, obtaining the set denoted byQi+u0.

2. Morphological erosion of setQi+u0 by ξ0, obtaining the set denoted by Qi+ u0−ξ0.

3. Let Qi+1 be the points q of Qi, for which f(q) is inside the set denoted Qi + u0−ξ0.

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14 Chapter 2. Description of the partial control method

Figure 2.2. Graphical process used by the Sculpting Algorithm to obtain

the safe set. The denoted set Qi is fattened by the thickness u0. The fattened set is

displayed in red. Then, the new set is shrunk or contracted by a distance ξ0, obtaining

the set denoted Qi+u0−ξ0 (in green). Finally we remove the grid points q∈Qi whose

imagef(q) falls outside Qi+u0−ξ0. Notice thatQi+1 ⊂Qi.

this final region, the safe set. Note that if the chosenu0 is too small, thenQ∞

may be the empty set, so that a bigger value of u0 is required to control the

trajectories.

The computation of the safe set Q∞ requires to take a finite grid covering Q0,

since it is not possible to compute the infinite number of points in Q0. We will call

the grid resolution as the distance between two adjacent pointsq. Higher resolutions give a more accurate safe set, and beyond a critical resolution of the grid covering

Q and ξ, the safe set remains practically unchanged. Due to the complex shape of the chaotic saddle underlying the chaotic dynamics, the derivation of a rigorous proof of the convergence of the algorithm would be extremely difficult. However, from a computational view, it is easy to show that the algorithm converges in a finite number of steps, since the grid used is composed of a finite amount of points. From a practical point of view, we recommend to compute the safe set with the algorithm proposed with increasing resolutions until finding the critical value for which the shape of the safe set found remains unchanged. That one will be a very good approximation of the real safe set. We also should take into account that the finite resolution of the computation is by itself a source of disturbance, so the magnitude of the disturbance can never be zero.

In Fig. 2.3 it is shown an example of a safe set that was computed for the Lorenz system for a choice of parameters where transient chaos is present. Without control, trajectories eventually escape toward two external fixed points. The points belonging to the safe set, can be controlled and kept in the chaotic region forever. In Chapter 4, this system is studied in detail.

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2.2. The partial control method 15

Figure 2.3. Example of the set needed to partially control the Lorenz system.

The figure shows an example of a 3D safe set in the phase space computed for the partially controlled Lorenz system in the transient chaotic regime. The blue set represents the points of the phase space that satisfy the control condition defined by the partial control method. The red set is a subset of the blue set, and represents the asymptotic region where the controlled dynamics converges.

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Chapter 3

Partial control to avoid a

species extinction

Extinction of species is one of the most dramatic processes in ecology. In case where extinction depends on the population dynamics of other species, to avoid the extinction might is a big challenge from an ecological point of view. The nonlin-ear interactions among species together with the presence of noise, often result in a complex global dynamics, making difficult to predict external actions over the system.

Here we use an extended version of the McCann-Yodzis [18] three-species food chain model proposed by Duarte et al. [27], where a cooperative hunting term was added to the original McCann-Yodzis model and where the three species coexist: resources, consumers and predators. We consider a situation for which a chaotic transient is present in the dynamics implying the predators extinction. Taking into account that the system is affected by external disturbances, we implement the partial control with the goal of avoiding the extinction. We have also shown that the partial control method implies smaller controls.

3.1

Partial control to avoid a species extinction

In this application of the partial control method, we have worked with an ecological model that describes the interaction between three species: resources, consumers and predators. The interest of this model lies in the fact that, for some choices of parameters, transient chaos appears involving the extinction of one of the species, the predators. Without no control, the system evolves from a situation where the three species coexist towards a limit cycle where just resources and consumers survive.

The model that we have used is an extension of the McCann-Yodzis model [18] proposed by Duarte et al. [27], which describes the dynamics of the population density of a resource species R, a consumer C and a predator P. The resulting model is given by the following set of nonlinear differential equations:

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18 Chapter 3. Partial control to avoid a species extinction

Figure 3.1. Dynamics of the extended McCann-Yodzis model proposed by

Duarte et al. from Eqs. (3.1). Depending on the values of the parameters (K, σ)

different dynamics are possible. Fixing K = 0.99, the boundary crisis appears at σc =

0.04166. (a) Before the boundary crisis (K = 0.99,σ= 0), there are two possible attractors depending on the initial conditions: one chaotic attractor where the three species coexist, and one limit cycle where only the resources and consumers coexist. (b) After the boundary crisis (K = 0.99, σ = 0.07), the limit cycle is the only asymptotic attractor. (c) Time series of the predators population corresponding to the case (b), where the chaotic transient before the extinction is shown.

dR dt =R

1− R K

−xcycCR R+R0 dC

dt =xcC

ycR R+R0

−1

−ψ(P) ypC

C+C0

(3.1)

dP

dt =ψ(P) ypC C+C0

−xpP.

Note that R, C and P are non-dimensional variables. Following [18] and [27], we have fixed the ecological parameters: xc = 0.4, yc = 2.009, xp = 0.08, yp = 2.876,

R0 = 0.16129, C0 = 0.5, K = 0.99 and σ = 0.07. For these values transient chaos

behaviour appears, and the predators eventually get extinct (see Fig 3.1).

With the aim of avoiding the extinction, we have applied the partial control method [16]. Since this model is a flow, we need first to discretize the dynamics to built a map. Different choices are possible. In this case we have chosen to build the map by using the successive local minima (Pn, Pn+1), wherePndenotes thenth local

minimum of theP variable time series. This set of points generates an approximately one-dimensional curve in the phase space as shown in Fig. 3.2. The corresponding return map of the formPn+1 =f(Pn) is shown in Fig. 3.3. Notice that, the iterates

of any initial point for which Pn > P∗

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3.1. Partial control to avoid a species extinction 19

Figure 3.2. Set of local minima in phase space. Blue line: trajectories in phase plane (P, C) and (P, R). Green line: set of local minima of P(t) used to build the one-dimensional map. As shown, the set of local minima is approximately parallel to the P

axis.

the extinction of the predators population. We assume that the map constructed in this way is just an approximation, so we also introduce a disturbance term ξn into the map, in order to model potential mismatches.

After introducing the disturbance term ξn and the control term un in the map, the partially controlled dynamics is given by:

Pn+1 =f(Pn) +ξn+un. (3.2)

In our case, we want to sustain the dynamics close to the chaotic attractor, avoiding the escape produced whenPn < P∗ = 0.589, therefore we choose the initial

Q region to be the interval Pn∈[0.589,0.84] indicated in Fig. 3.3. Then we use the Sculpting Algorithm to find the safe set.

The computation of the safe set depends on the chosen values of ξ0 and u0. To

show an example, we have chosen for our simulations the values ξ0 = 0.0114 and u0 = 0.0076, where u0 is very close to the minimum value for which the safe set

exists. In Fig. 3.3 we represent the steps of the algorithm to build the safe set. In Fig. 3.4, we represent the obtained final safe set that allows us to control the map constructed with the minima of the P variable. Notice that from the point of view of the real dynamics (continuous trajectories), the control is applied every time the trajectory crosses the set of minima. If the value f(Pn) +ξn is inside a safe set, we do not apply control, and if it is outside, we relocate it inside the nearest safe point, resulting the new safe point Pn+1=f(Pn) +ξn+un. The criterion to control

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20 Chapter 3. Partial control to avoid a species extinction

Figure 3.3. Return map Pn+1 = f(Pn) obtained by using the successive local

minima of the time series P(t). Notice that below P∗ = 0.589 the trajectory

asymp-totes to zero. In order to keep the trajectory in the region Q indicated, we compute the safe set. In the lower part are shown the steps of the Sculpting Algorithm that converges to the final safe set. The horizontal black bars helps us to visualize the process and rep-resent the points Pn that satisfy the condition to be a safe point at each step. In this

case, the upper bound of disturbance and control used areξ0 = 0.0114 and u0 = 0.0076,

respectively.

In Fig. 3.4 it is also represented the asymptotic safe set. This subset of the safe set [17] appears typically when the system is dissipative, and represents the asymptotic region of the safe set where the controlled trajectories converge. Once the trajectories enter into the asymptotic safe set, they never abandon it.

Controlled trajectories in phase space are shown in Fig. 3.5, where we also in-dicate the safe set used with the projections on the set of the minima for a clear visualization. In Fig. 3.6, we represent the corresponding controlled time series of the predators population (blue line) in contrast to the uncontrolled trajectory (red line), involving the extinction. On the right of the figure we also represent a zoom of one minimum to highlight how the noise (that we call disturbance) appears and how the control is applied. Notice that the noise amplitude shown only represents the difference between the deterministic trajectory and the noisy one.

In order to see a further analysis, in Fig. 3.7 it is represented the strength of the noise and the strength of the control applied for 30.000 iterations corresponding to the time interval [0,1.2×106

] in the time series of P(t). While the strength of the noise (absolute value) is distributed uniformly between the values 0 andξ0 = 0.0114,

all values of the control (absolute value) are located under the maximumu0 = 0.0076,

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3.1. Partial control to avoid a species extinction 21

Figure 3.4. Final safe set. The safe set is composed of different subsets obtained with the Sculpting Algorithm using ξ0 = 0.0114 and u0 = 0.0076. We also indicate the group

of subsets where the dynamics remains trapped, that is, the asymptotic safe set.

Figure 3.5. Controlled trajectory in the phase space. Controlled trajectory with

ξ0 = 0.0114, represented in the phase plane (P, C). We also show the asymptotic safe

set computed with ξ0 = 0.0114 and u0 = 0.0076, and its projection in the set of minima

(dashed line) where the control is applied.

control that we need to use, is another remarkable feature of this control method. To show how the safe sets change depending on the upper bound value of noise

ξ0, we represent in Fig. 3.8 different safe sets with ξ0 in the range [0.001,0.057], and

the corresponding controlled trajectories in phase space. Note that the trajectories are sustained in the chaotic region with |un| ≤ u0 for all the iterations, and the

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22 Chapter 3. Partial control to avoid a species extinction

Figure 3.6. Controlled time series. Red line: Time series of the predators population without control exhibiting a escape towards zero, that implies the extinction of the preda-tors. Blue line: Controlled time series of the predators population where the extinction is avoided. At every minimum, the value of P is evaluated and if necessary a small control is applied. This time series corresponds to 50 iterations in the return map Pn+1 =f(Pn).

A zoom of one of the minima of the time series ofP(t) is also shown on the right in order to see how the noise is introduced and how the corresponding control is applied.

Figure 3.7. Strengths of noise and control applied for 30000 iterations. Noise and control amplitude are represented as points instead of bars for a clear visualization. On the left, the strength of the noise that affects the map. On the right, the respective strength of controls applied for the partial control method. We also indicate the upper bound of the noise ξ0 = 0.0114 and upper bound of the control u0 = 0.0076 used to

compute the safe set.

3.2

The partial control method implies smaller controls

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3.2. The partial control method implies smaller controls 23

0,55 0,7 0,85 1

0.1 0.27 0.44 0.6

Safe Set ( ξ

0=0.0020 , u0=0.0017 )

P

C

Average noise =0.0010 Average control=0.0003

(a)

0,55 0,7 0,85 1

0.1 0.27 0.44 0.6

Safe Set ( ξ

0=0.010 , u0=0.007 )

P

C

Average noise =0.0052 Average control=0.0016

(b)

0,55 0,7 0,85 1

0.1 0.27 0.44 0.6

Safe Set ( ξ

0=0.50 , u0=0.32 )

P

C

Average noise =0.026 Average control=0.011

(c)

Figure 3.8. Different disturbance magnitude. Controlled trajectories in the phase plane (P, C) and the respective asymptotic safe sets for different noise intensities in the range ξ0= [0.002,0.50].

it does not contemplate at all the presence of any external disturbance.

The main idea here is to consider the problem analyzed in [27], including an external disturbance, and compare both methods, the Dhamala and Lai strategy and the partial control.

The control method of Dhamala and Lai is based on the observation that a point

P < P∗ in the return map, goes quickly to zero. In this sense, it is possible to

identify one escaping zone and two target regions, by computing certain preimages of the fixed point P∗

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24 Chapter 3. Partial control to avoid a species extinction

Figure 3.9. Dhamala and Lai method applied to the map. Theblack setsrepresent the two target regions used to control the trajectories. The red sets represent the escape regions. We colored the regions with different thicknesses to help us in the visualization.

region, survive a long time before escaping.

The control is defined as follows: we have two target regions (black sets) defined by the intervals [a1, b1] and [b1, c1]. Now we define the escaping regions (red sets),

composed by the points P < P∗ and the points between the target regions, see

Fig. 3.9. When a given iteration falls into the escaping region, we apply a control to relocate P inside the nearest target point. Note that, in contrast with safe sets of the partial control that are different depending on the bounds of disturbance and control, the target sets of the Dhamala and Lai method are always the same.

In Fig 3.10, we represent the average control applied in both methods for different values of the strength of the disturbance in the range ξ0 = [10−3,10−1]. In view of

the results, we can say that the Dhamala and Lai control strategy, implies larger controls than the partial control method in almost all situations.

One could think that this different performance is due to the presence of distur-bance, and the fact that the Dhamala and Lai method was not designed to deal with disturbances, but in fact, the smaller the disturbance the better results achieves the partial control. In contrast, when the disturbance is too large, both methods achieve similar results. The reason is because large disturbances blur the chaotic saddle so much, that only a gross control is able to avoid the escape.

3.3

Conclusions

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3.3. Conclusions 25

Dhamala and Laia

b

Figure 3.10. Dhamala and Lai method vs partial control method. Comparative of the average controls in a log-log scale. Dashed black line: Average strength of the noise in the rangeξ0 = [10−3,10−1]. (a) Red line: Average strength of the control method used

in [27]. (b) Blue line: Average strength of the partial control.

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Chapter 4

Controlling chaos in the

Lorenz system

The Lorenz system is a paradigmatic example in Nonlinear Dynamics that, ex-hibits transient chaos for a certain choice of parameters. In this regime trajectories eventually converge to two fixed points attractors. In this chapter, we analyze three quite different ways to implement the partial control method, in order to avoid escapes. First, we apply this method by building a 1D map using the successive maxima of one of the variables. Next, we implement it by building a 2D map by using a Poincar´e section. Finally, we built a 3D map, which has the advantage of using a fixed time interval between the application of the control, which can be useful for practical applications.

4.1

Partial control to avoid the fixed point attractors in the

Lorenz system

The Lorenz system [28] is a flow that describes a simplified model of atmospheric convection. The model consists of three ordinary differential equations,

˙

x=−σx+σy

˙

y=−xz+rx−y (4.1)

˙

z =xy−bz.

Depending on the parameter values r, σ, and b, the system can exhibit different dynamical behaviors, either periodic solutions, chaotic attractors or even transient chaos. Fixing σ = 10, b = 8/3, transient chaos can be found in the interval r ∈

[13.93,24.06] as described in [29, 30]. For our simulations, we have chosen the value

r= 20.0. In this regime, as we show in Fig. 4.1, there are transient chaotic orbits that eventually decay towards one of the two point attractors which physically represent a steady rotation of a fluid flow, one clockwise, and the other counterclockwise. The point attractors are located in the following positions,

C+

= (pb(r−1),pb(r−1), r−1)≈(7.12,7.12,19)

C−

= (−pb(r−1),−pb(r−1), r−1)≈(−7.12,−7.12,19).

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28 Chapter 4. Controlling chaos in the Lorenz system

Figure 4.1. Dynamics of the Lorenz system. We select the transient chaotic regime withσ= 10,b= 8/3 andr= 20. On the left, the trajectory is deterministic. On the right, the trajectory is affected by some disturbances. The disturbances here, were enlarged in order to help the eye. Almost all trajectories eventually spiral to one of the two attractors (C+

or C−). Here both trajectories spiral toC+

.

In this figure, we also represent the case where some noise is present in the trajectory. The noisy trajectory behaves similarly to the deterministic one. The main difference is the time involved to reach the attractors, which can be increased or reduced.

The goal of applying control here is to avoid trajectories falling in one of the attractors. To apply the control method, we need first to built a map, however we have found many interesting possibilities. Here we explore three of them, consisting of a 1D, 2D and 3D maps respectively.

4.1.1

The 1D map

As shown by Lorenz [28], when plotting the pairs of maxima (zn, zn+1), one gets

(approximately) a functionf wherezn+1 ≈f(zn). We can see this clearly in Fig. 4.2.

Knowing a local maximum of z isZ, allows one to estimate |x|and |y|with consid-erable precision. Transient chaos can be observed in the intervalzn ∈[27.3,30.7], so we have chosen this interval as the set Q0. We have taken ξ0 = 0.080 and the

con-trol bound u0 = 0.055 (u0 < ξ0). This control value is approximately the minimum

value for which a safe set exists. Then, we have obtained the safe set by using the Sculpting Algorithm. In Fig. 4.2, we can see how the algorithm sculpts the initial region Q0 until it finds Q4 where it converges, so Q4 =Q∞ is the safe set. For this

computation we have used a grid of 4000 points in the interval zn ∈ [26.8,30.8], so the grid resolution is 0.001.

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4.1. Partial control to avoid the fixed point attractors in the Lorenz system 29

Figure 4.2. The 1D safe set. The black curve is the 1D map built with the successive maxima ofz. We take as initial setQ0 (upper segment in blue) the region where transient

chaos occurs. The map is affected by disturbances with an upper boundξ0= 0.080, while

we choose the upper bound of the control as u0 = 0.055, (the bounds are the width of

the bars displayed in the upper left side). The figure shows the successive steps computed by the Sculpting Algorithm, from an initial region Q0 until it converges to the subset Q4 = Q∞ ⊂ Q0. We use a grid of 4000 points in the interval zn ∈ [26.8,30.8], that

corresponds to a resolution of 0.001.

coordinates (xm and ym) as in the case of the ecological model treated in the pre-vious chapter where we have applied control to one of the variables (the predators species). The main advantage of this 1D map is that the computation of the safe set is easy and fast. This kind of map is useful when the main component of the disturbance affects the variable we use to built the map.

4.1.2

The 2D map

It is straightforward to build a 2D map taking a Poincar´e section that intersects the flow. For our purpose, we have chosen the plane z = 19 with the ranges x ∈

[−3,3] and y ∈[−3,3], as shown in Fig. 4.4. The trajectories that cross this plane are in the transient chaotic regime, while the attractors C+

= (7.12,7.12,19) and

C−= (7.12,7.12,19) that we want to avoid, are situated outside this plane (see

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30 Chapter 4. Controlling chaos in the Lorenz system

Figure 4.3. Time series of the variable z for the Lorenz system with r=20.

The figure shows a comparison between an uncontrolled trajectory that escapes from chaos (red line) and a partially controlled trajectory (black line). Starting with the same initial

condition, the uncontrolled trajectory eventually decays to C+

or C−, which physically means a steady rotation of the fluid flow. On the other hand the partially controlled trajectory is maintained in the chaotic transient regime, that is, the rotation of the fluid flow remains chaotic forever.

z = 19. Then we have used the Sculpting Algorithm to find the safe set Q∞⊂Q.

As an example, we have assumed that the map is affected by some disturbances with upper bound ξ0 = 0.09. The minimum control found for which the safe set

exists is u0 = 0.06. In Fig. 4.5(a), this safe set is displayed. A partially controlled

trajectory is represented in Fig. 4.5(b), where we have also shown the safe set in phase space in order to see how it is used to control the system. Notice that, we are able to avoid the attractors, applying only small perturbations in the plane. A zoom of this region is shown in Fig. 4.5(c). The computation was carried out taking a grid size of 1200×1200 points, (grid resolution is 0.005 in both variables x and

y).

The main advantage of the 2D map is that allows to partially control systems where all the variables are affected by disturbances since the imagexn+1 =f(xn)+ξn

in the Poincar´e surface is a certain ellipse, and both dimensions of the surface are controlled. In addition, as opposed to the 1D map, where we have to act on the x,

y and z variables to control the system, the control in the 2D map is only applied in the variables x and y, since z is constant. This can be an advantage in systems where it is difficult or expensive to apply the control in each variable.

4.1.3

The 3D map

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4.1. Partial control to avoid the fixed point attractors in the Lorenz system 31

Figure 4.4. The Lorenz system with r=20 (transient chaos). The figure shows an uncontrolled trajectory in phase space crossing the square withx∈[−3,3] andy∈[−3,3] in the plane z = 19. To built the map, we use a grid of initial conditions in the plane, and evaluate the images of the trajectories when they cross again the plane. The goal of the control will be to keep the trajectories in this plane, avoiding the escape to one of the attractors C+ or C, placed outside.

x(t0), y(t0), z(t0) → x(t0+ ∆t), y(t0 + ∆t), z(t0 + ∆t). By computing the time-∆t

image of each point of a 3D grid that covers the phase space, we can obtain the 3D map. We discuss the advantages of this map below.

To built this kind of map, a suitable choice of ∆t is important. For too small values, safe sets (with u0 < ξ0) does not exist. The topological explanation for this,

is that the flow is acting like a pastry transformation which takes some time to be completed. Once this time is reached, the safe set appears. For our Lorenz system, there are safe sets for values of ∆t≥1.2.

For the computation of the safe sets, we consider the domain with x∈[−20,20],

y∈ [−20,20], z ∈ [0,40], with a grid size of 400×400×400, so the grid resolution is 0.1 for each variable. In this region the transient chaotic trajectories eventually decay to the attractors C+

= (7.12,7.12,19) andC− = (7.12,7.12,19). In order

to avoidC+

andC−, balls centered in these attractors are removed. See the regionQ

and a transient chaotic trajectory in Fig. 4.6. To obtain the map, we have computed the image of each point ofQwith ∆t= 1.2. Then, as an example, we have taken the valueξ0 = 1.5 andu0 = 1.0 (noteu0 < ξ0). After applying the Sculpting Algorithm,

the safe set shown in Fig. 4.7(a) is obtained.

To describe the controlled dynamics in the 3D map we writeqnfor the controlled trajectory at timen∆t. To obtain a particular trajectory, we start with a given state

qn and then we compute the image qn+1 = f(qn) +ξn+un, where ξn is chosen at

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32 Chapter 4. Controlling chaos in the Lorenz system

(a) Safe set (b) Safe set and controlled tra-jectory

(c) Zoom of the controlled tra-jectory

Figure 4.5. The 2D safe set and how it is used to control the trajectory. (a) The safe set obtained using the map built with the plane displayed in Fig. 4.4. We show in blue the computed safe set Q∞ for ξ0 = 0.09 and u0 = 0.06 (u0 < ξ0). The grid size

used is 1201×1201 points. The radius of the balls in the lower left side indicates the bounds of the disturbance, ξ0 = 0.09 (green) and the control u0 = 0.06 (yellow). (b) A

partially controlled trajectory in phase space. Each time that the trajectory crosses the safe set plane (placed in z = 19), the control is applied pushing the trajectory onto the set avoiding the escape from chaos. (c) Zoom of how the control is applied in the safe set.

safe set. In each case, ξn represents the disturbance accumulated by the trajectory in the time interval ∆t= 1.2, while the control is always applied at a discrete time. In this case, we apply the minimum control, however other criterion is possible as long as the constraint |un| ≤u0 is respected.

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4.1. Partial control to avoid the fixed point attractors in the Lorenz system 33

Figure 4.6. A choice of 3D set Q. The 3D set Q is the cube x ∈ [−20,20], y ∈

[−20,20], z ∈ [0,40] except the balls of radius 4, centered in C+

= (7.12,7.12,19) and

C−= (7.12,7.12,19) that are removed fromQ. We want trajectories to stay inQ and not fall to these attractors. A trajectory is plotted to show the chaotic transient behavior in this region.

The controls, represented as yellow segments distributed along the trajectory, are applied every ∆t = 1.2. We show this fact with a zoom in Fig. 4.7(d). As a result, the trajectories never fall into the attractorsC+

orC−, keeping the dynamics in the

chaotic region forever.

As we have mentioned, the safe set appears for values of ∆t ≥1.2, so it is possible to adapt the control frequency to our specific requirements, taking longer ∆t values. To show that, we compute in Figure 4.8(a) the asymptotic safe set for ∆t = 1.8 , and with ξ0 and u0 unchanged. With this set we could control the system applying

a control every ∆t = 1.8 (see Fig. 4.8(b)) instead of ∆t = 1.2 as in the previous case. However taking a longer ∆t has a downside since in most scenarios that the cumulative effect of disturbances grows exponentially with time due to chaos, and therefore it is expected that the bound of control u0 needed increases as well.

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34 Chapter 4. Controlling chaos in the Lorenz system

(a) Safe set with ∆t= 1.2 (b) Asymptotic safe set

(c) Controlled trajectory (d) Zoom of the controlled trajec-tory

Figure 4.7. The 3D safe set and how it is used to control the trajectory. (a) In blue the 3D safe set Q∞ for Fig. 4.6, obtained after applying the Sculpting Algorithm. We set ∆t = 1.2, ξ0 = 1.5 (ξ0 = radius of the green ball) and u0 = 1.0 (u0 = yellow

ball’s radius). In red the asymptotic safe set which is a subset of the safe set. This is the region in which the controlled trajectories eventually lie. (b) The asymptotic safe set alone. Partially controlled trajectories converge rapidly to this region. (c) A cut-away section of the asymptotic safe set in order to see a partially controlled trajectory (with ∆t = 1.2) displayed in black. The controls (yellow segments inserted in the trajectory) are applied every ∆t= 1.2. As a result, the trajectory is kept in the chaotic region and the attractors

C+

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4.2. Conclusions 35

(a) Safe set with ∆t= 1.8 (b) Asymptotic safe set

Figure 4.8. Different safe set for different values of ∆time. (a) The asymptotic safe set computed for ∆t= 1.8. To compute this set we have taken ξ0 = 1.5 (green ball)

andu0 = 1.0 (yellow ball).(b) A half section of the asymptotic safe set (red) and a partially

controlled trajectory (in black). In this case the controls (yellow segments inserted in the trajectory) are applied every ∆t= 1.8.

Figure 4.9. Comparison of the three controlled trajectories of the z variable

obtained with the 3D, 2D and 1D map respectively. The marks indicate the

moment where the control is applied. Only in the 3D case are the controls time periodic.

4.2

Conclusions

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36 Chapter 4. Controlling chaos in the Lorenz system

ways. We have built 1D, 2D and 3D maps, and obtained the respective safe sets with the Sculpting Algorithm.

Using the respective safe sets in each case, we have shown that is possible to control the trajectories, using a small amount of control in comparison with the disturbances affecting the system. Another remarkable feature is that the partially controlled trajectories keep the chaotic behavior of the original system.

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Chapter 5

A different application of

partial control

There are certain situations in noisy nonlinear dynamical systems, where it is required a fast transition between a chaotic and a periodic state. Here, we present a novel procedure to achieve this goal in the context of the partial control method of chaotic systems. We will show that, by only using the safe sets it is possible to handle the stabilization and destabilization of the chaotic dynamics of the partially controlled system.

5.1

Safe set to avoid the escape or force it

In contrast with the previous chapters where the partial control method was used to keep trajectories close to the chaotic saddle and to avoid an undesirable escape [31, 32, 33, 34], the goal here is to maintain the chaotic transient as much as we desire, before forcing an immediate escape. To do that, we use the same safe sets defined in the partial control method in a completely different way.

It is reasonable to think that, in the case of a transient chaotic dynamics, the simplest strategy to force the escape of the trajectories is just to stop applying the control and wait until the trajectory naturally escapes. However, in many cases the average time between the moment in which the application of the control is stopped and the moment in which the trajectory reaches the escape may be very long. It is here where we found that the safe set can be used in a different way to speed up the escape time of the trajectory and therefore to get a higher control in the behaviour of the system. We show here that a practical way to achieve this goal is simply to apply the control to drive the trajectories outside the safe set. This strategy is supported by the fact that non-safe points usually have longer escape times than the safe points. In addition points far from the safe set typically have longer escape times.

The strategy to accelerate the escape of the trajectory, consist on applying a control |un| ≤ u0 each iteration to the most far away point of the safe set. As we

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